Analysis of traveling waves for nonlinear degenerate viscosity of chemotaxis model under general perturbations

Abstract

In this paper, we generalized the results of the following chemotaxis model with the nonlinear degenerate viscosity $ \begin{equation*} \begin{cases} u_{t} -\chi (uv)_{x} = D(u^{m})_{xx}, \\ v_{t} -u_{x} = 0, \end{cases} \end{equation*} $ by introducing the following general initial perturbation $ \begin{equation*} \begin{split} \int_{-\infty}^{+\infty}\kappa(Z_0|\tilde{Z})dx<\infty, \end{split} \end{equation*} $ where $ \kappa $ is the relative entropy function defined in Eq (2.24). We further employed the relative entropy method by choosing the specific shift function. According to the estimates with the cutoff version, and overcoming the complexity caused by the porous media diffusion, the nonlinear orbital stability of traveling waves was established under small amplitude and general perturbations.

Keywords

• chemotaxis
• nonlinear diffusion
• orbital stability
• energy estimates cutoff
• general perturbations